Definition df-ms 12509

Description: Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)

Assertion

Ref	Expression
df-ms	⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

Detailed syntax breakdown of Definition df-ms

Step	Hyp	Ref	Expression
1		cms 12506	. 2 class MetSp
2		vf	. . . . . . 7 setvar 𝑓
3	2	cv 1330	. . . . . 6 class 𝑓
4		cds 12030	. . . . . 6 class dist
5	3, 4	cfv 5123	. . . . 5 class (dist‘𝑓)
6		cbs 11959	. . . . . . 7 class Base
7	3, 6	cfv 5123	. . . . . 6 class (Base‘𝑓)
8	7, 7	cxp 4537	. . . . 5 class ((Base‘𝑓) × (Base‘𝑓))
9	5, 8	cres 4541	. . . 4 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
10		cmet 12150	. . . . 5 class Met
11	7, 10	cfv 5123	. . . 4 class (Met‘(Base‘𝑓))
12	9, 11	wcel 1480	. . 3 wff ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))
13		cxms 12505	. . 3 class ∞MetSp
14	12, 2, 13	crab 2420	. 2 class {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
15	1, 14	wceq 1331	1 wff MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

This definition is referenced by:  isms  12622